Decomposing Feynman rules

TL;DR

This paper analyzes the structure of parametric renormalization in Feynman graphs, demonstrating a decomposition into angles and scales using a simple example to clarify the process.

Contribution

It introduces a decomposition of renormalized Feynman rules into angle and scale components at the integrand level, enhancing understanding of renormalization structure.

Findings

Decomposition of Feynman rules into angles and scales.

Explicit example with Dunce's cap graph.

Clarification of parametric renormalization process.

Abstract

We exhibit in the simple example of the Dunce's cap Feynman graph the structure of parametric renormalization on the level of integrands, and exhibit also the decomposition $\Phi^R=\Phi_{\mathrm{fin}}^{-1}(\Theta_0)\star \Phi^R_{\mathrm{\makebox{1-s}}}(S/S_0)\star\Phi_{\mathrm{fin}}(\Theta)$ into angles and scales of renormalized Feynman rules.